Defining Practice

Overview

Defining, a central practice in mathematics, is different from the daily practice of clarifying the meaning of words. Mathematics students often have trouble formulating definitions, interpreting new definitions, and using definitions in proof and problem solving. And whereas students may be able to state many mathematical definitions, these are often presented and received as facts. The perspective taken here is that mathematical definitions are not inherent truisms but rather meaning-relations built to fulfill particular functions, often via a community's iterative process of formulating and evaluating a definition with respect to its utility, conformity to intuition, or other criteria.

The primary goal of this design-based research project is to explore the semiotic strategies students use, develop, and negotiate when they engage in solving problems involving interpersonal specificity of information, such as in responding to a situation of apparent referential ambiguity. These spontaneous practices of generating mathematical definitions as pragmatic solutions to repair problems of communication may differ from students’ descriptions of their processes of formal defining. As such, a secondary goal of this project is to investigate how the naturalistic discursive practices of repairing referential ambiguity could be leveraged in creating pedagogical opportunities to engage the disciplinary practice of generating formal definitions. Central to the current study is a learning design called the Specification Game.

The Specficiation Game: A game board (left) and associated game pieces (right).